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How do I solve the equation x^7 \equiv 21 modulo 66

66 = 2.3.11 so try solving it mod 3 and mod 11 (mod 2 doesn't gives any new information). This tells us x is divisible mod 3, and x^7 = -1 mod 11. One solution to this is x = -1 mod 7. The smallest solution of these two congruences is x=21, so this seems like a good guess.

Now notice that 21*21 = 3.7 (2.11 - 1) = 7.66 - 3.7 = -21 mod 66

So 21 ^n = 21. (-1)^n mod 66. Therefore x=21 is a solution of the equation. There may be more however...

Sincerely Yours

Hummingbird25

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